Convexity and some geometric properties

TL;DR

This paper investigates the existence and properties of convex functions on Riemannian manifolds, linking their existence to geometric features like geodesic flow and curvature, and providing conditions for their construction.

Contribution

It establishes new geometric conditions for the existence of convex functions on non-compact manifolds and relates convexity to manifold curvature and geodesic flow.

Findings

Convex functions do not exist on manifolds with infinite volume and conservative geodesic flow.

A geometric condition guarantees the existence of convex functions on certain complete non-compact manifolds.

Manifolds with sectional curvature greater than a negative constant can admit strictly convex functions.

Abstract

The main goal of this paper is to present results of existence and non-existence of convex functions on Riemannian manifolds and, in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Rieman- nain manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete non-compact man- ifolds, and, we use this fact to construct a manifold whose sectional curvature assumes any real value greater than a negative constant and admits a strictly convex function. In the last result we relate the geometry of a Riemannian manifold of positive sectional curvature with the set of minimum points of a convex function defined on the manifold.